Always use half reactions:
Reduction:
HNO3 {oxidation nr. of N = 5} + H2O + 3e- --> NO {oxidation nr. of N = 2} + 3OH-
difference in oxidation nr is 3, hence 3 electron reaction
Oxidation:
P {oxidation nr. = 0} + 5 OH- --> H3PO4 {oxidation nr. of P = 5} + H2O + 5 e-
difference in oxidation nr. is 5, hence 5 electron reaction.
Note: a half reaction is only balanced if the charges are also balanced. Electrons are a necessary part of a redox half-reaction
Water/OH- is just used to balance O and H (water+ H+ is also good to balance of course)
after that just balance the two and you will indeed see the answer is E.